558 Mr. Gr. H. Livens on the Electron 



Remembering the proper significance to be attached to F , 

 we see at once that this result is, to the order of approxi- 

 mation adopted, identical with that given above and obtained 

 by Lorentz's method. The two methods thus consistently 

 provide the same results in a simple and straightforward 

 manner in the most general possible case of the type under 

 consideration. 



4. The basis oj the Jeans- Wilson-Bohr theory. 



Before concluding this account it may serve some purpose 

 to give a short discussion of the fundamental basis of the 

 modern extensions of the theory originated by Jeans and 

 developed by Wilson and Bohr, at least so far as concerns 

 its connexion with the methods of the foregoing paragraphs. 

 The new method of attack proposed by Wilson and corrected 

 and completed by Bohr, consists in the determination of an 

 equation specifying the rate of variation of the aggregate 

 momentum of the group of all the electrons per unit volume 

 which have their absolute velocities (u) within a specified 

 small range between u and u + du. The equation obtained 

 by Bohr can easily be deduced from the results obtained 

 above and in the following manner. The differential equation 

 satisfied by the function /is 



d£ ^v o? ox oy oz & t« 

 We may now remark that the function / differs but 

 slightly from / and by a term which is linear in (X, Y, Z) 

 and the condition gradients in the metal, which quantities 

 are assumed to be such that squares and products of them 

 are negligible under all circumstances. It is, therefore, 

 quite indifferent whether we use / or f in the first six 

 terms of the left-band side of this equation. Now if for 

 simplicity we assume, for instance, that 



then 





^v df d.o dy b d- 





=/o[K-^+-i-^-4!) 





-(-^4t--g) 





+ <-^ + i£--g)] 



